A professor at a University X is holding a second appointment as a researcher on leave at another University in a different country with his full salary being paid from his funding there. The professor in question is likely being employed and paid as a full-time assistant professor at uni X despite rarely showing up there which may be illegal.
How can I check if a professor who holds a second appointment as a faculty researcher and is being paid full time in both appoinments?
The only reasonable way to know this information is to ask him, though it seems unlikely he will answer. You could also obtain this information through unreasonable or unethical means such as gossip, espionage, or theft but these are not advisable. Be aware you may be violating the privacy of this individual.
You suggest that this activity may be illegal. Legality will depend on employment law in your jurisdiction. I am not sure from your question if you have admissible evidence to make this claim. If you believe illegal activity has been committed, you may report it to HR, institutions which handle employment oversight, or even the police. They may not be able to pursue it without first being provided evidence. HR may be the best bet in the case that you do not have evidence, since they might be motivated to obtain evidence. It may be advisable to make any report anonymously as a whistleblower in order to minimize your exposure.
Why does it matter to you? The details of employment are between the employer and employee. If both sides (or in this case all three sides) are happy with the arrangement, it is nobody else's business figuring out who pays whom and how much. If one of the two universities are unhappy with the situation, it is their job (not yours) to make inquiries.
You don't get to check anything. The arrangements made between that professor and these universities are none of your business.
If you are really really worried, you could give what you know to the HR department of your university. However, assuming they are professional, they will not tell you anything. They may or may not already know what you know. They may or may not have an arrangement in place. They may or may not take action against it. But they will tell you nothing, because it is between them and the professor.
Basically, it boils down to whether OP has a vested interest in the matter.
This can be as responsible officer or as bystander interested in the integrity of the process.
However, the "behind-the-scenes" information that OP gives makes them look like they are looking for material to create damage to the target person as punishment for other wrongdoings. Anything but initiating investigation and possible punishment for the specific misdeed in question is vigilantism and morally very problematic.
Cross-punishment in civilized societies is not permitted for very good reasons.
Concretely, if the person in question did plagiarize or other academic misdeeds, these should be reported as such to the relevant entities. If the person has a fraudulent employment status, that may not be OP's business, but it's also OP's right as upright citizen to report that.
But what I found very questionable is that OP seems to intend to cross-retaliate against the alleged academic misdeeds of the prof using a completely unrelated employment matter. That would be ethically very questionable and you would be playing private avenger.
